What this circle calculator does
A circle is completely determined by any one of its four key measurements: radius, diameter, circumference, or area. This tool takes whichever one you know, computes the other three exactly, and shows every step. It also draws the circle to scale so you can see the actual size against a reference grid. Because π is irrational, most circle answers involve π — the calculator reports both the exact form (like 25π) and the decimal approximation to enough precision to satisfy any school or entrance-test requirement.
Circle formulas
Variable definitions
- r — radius, the distance from centre to edge.
- d — diameter, the distance across the circle through the centre.
- C — circumference, the distance around the circle.
- A — area, the space enclosed.
- π — the constant 3.141592653589793…
Worked examples
Example 1: Given radius r = 5
- Diameter: d = 2r = 2 × 5 = 10.
- Circumference: C = 2πr = 10π ≈ 31.4159.
- Area: A = πr² = 25π ≈ 78.5398.
Example 2: Given diameter d = 14
- Radius: r = d / 2 = 14 / 2 = 7.
- Circumference: C = πd = 14π ≈ 43.9823.
- Area: A = πd² / 4 = π × 196 / 4 = 49π ≈ 153.9380.
Example 3: Given circumference C = 62.8319 (which is 20π)
- Radius: r = C / (2π) = 62.8319 / 6.2832 ≈ 10.
- Diameter: d = 2r = 20.
- Area: A = πr² = 100π ≈ 314.1593.
Example 4: Given area A = 78.5398 (which is 25π)
- Radius: r = √(A / π) = √(78.5398 / 3.1416) = √25 = 5.
- Diameter: d = 2r = 10.
- Circumference: C = 2πr = 10π ≈ 31.4159.
Example 5: A wheel with radius 30 cm rolls once — how far does it travel?
- The distance travelled in one revolution equals the circumference.
- C = 2πr = 2π × 30 = 60π ≈ 188.4956 cm ≈ 1.885 m.
Where circles appear in Class 9 & 10
Circle geometry is one of the largest topics in the Punjab board syllabus, and the formulas here connect to it directly:
- Circle theorems — chord properties, tangents, inscribed angles, cyclic quadrilaterals.
- Sectors and arcs — area and arc length are fractions of the full circle.
- Physics — circular motion, angular velocity, and the period of a pendulum.
- Real-life problems — wheel rotations, area of circular fields, and design problems.
Common mistakes to avoid
- Confusing radius with diameter. The radius is from centre to edge; the diameter is across the whole circle. d = 2r, not r/2.
- Using the diameter where the radius belongs. A = πr², not πd². If you know the diameter, halve it first.
- Forgetting to square the radius for area. A = πr², not πr. Area grows with the square of the radius.
- Writing the wrong units. Circumference and diameter are in linear units (cm, m). Area is in square units (cm², m²). Always attach the correct exponent.
- Rounding π too early. Keep π as a symbol until the final step. Rounding to 3.14 too soon introduces avoidable error.
- Assuming a chord is a diameter. A diameter is the longest chord — every diameter is a chord, but not every chord is a diameter.
FAQ
What is pi?
Pi (π) is the ratio of a circle's circumference to its diameter. It is an irrational number, approximately 3.14159265358979. Every circle, no matter its size, has this same ratio, which is why π appears in both the circumference and area formulas.
How do I find the radius from the circumference?
Use r = C / (2π). Divide the circumference by 2π. For example, if C = 31.4159, then r = 31.4159 / 6.2832 ≈ 5.
How do I find the diameter from the area?
First find the radius using r = √(A / π), then double it: d = 2√(A / π). For A = 78.5398, r = √(78.5398 / 3.1416) = √25 = 5, so d = 10.
Why is the circumference 2πr and not πr?
Because C = πd and d = 2r. Substituting gives C = π × 2r = 2πr. The two formulas are equivalent — one uses the diameter, the other the radius.
Can a circle have negative radius?
No. A radius, diameter, circumference, or area is a physical measurement and must be positive (or zero for a degenerate point). This calculator rejects zero and negative values.
What is the difference between circumference and perimeter?
They are the same concept — the distance around the boundary of the shape. For a circle, we use the special word 'circumference' instead of 'perimeter'. For polygons, we say perimeter.
Related tools
Related Hira Academy resources
Circle formulas and theorems appear throughout the Class 9 and 10 geometry chapters. Revise with our free Class 9 notes and Class 10 notes.